Ampere-Maxwell Law

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Claimed by Maria Rivero


Ampere-Maxwell Law

James Maxwell discovered that a time-varying electric field could be accompanied by a magnetic field. He thought of this after Faraday discovered that a time-varying magnetic field was accompanied by an electric field.

The time rate that is used for this equation is given by the derivative of the electric flux with respect to time. Because The derivative of the flux gives current over epsilon nod, the derivative of the electric flux times epsilon nod will also have units of amperes.

Where B is the magnetic field dl is the change in path The sum of I is the sum of the charges inside the path.

Example

Pick a closed, rectangular path in the xz plane with a height h and a width w. Calculate the speed v of the slab.



Simple

Middling

Difficult

A Mathematical Model

In order to find the potential difference between two locations, we use this formula [math]\displaystyle{ dV = -\left(E_x*dx + E_y*dy + E_z*dz\right) }[/math], where E is the electric field with components in the x, y, and z directions. Delta x, y, and z are the components of final location minus to the components of the initial location.

A Computational Model

How do we visualize or predict using this topic. Consider embedding some vpython code here Teach hands-on with GlowScript

Simple Example

In this example, the electric field is equal to [math]\displaystyle{ E = \left(E_x, 0, 0\right) }[/math]. The initial location is A and the final location is C. In order to find the potential difference between A and C, we use [math]\displaystyle{ dV = V_C - V_A }[/math].

Since there are no y and z components of the electric field, the potential difference is [math]\displaystyle{ dV = -\left(E_x*\left(x_1 - 0\right) + 0*\left(-y_1 - 0\right) + 0*0\right) = -E_x*x_1 }[/math]

Let's say there is a location B at [math]\displaystyle{ \left(x_1, 0, 0\right) }[/math]. Now in order to find the potential difference between A and C, we need to find the potential difference between A and B and then between B and C.

The potential difference between A and B is [math]\displaystyle{ dV = V_B - V_A = -\left(E_x*\left(x_1 - 0\right) + 0*0 + 0*0\right) = -E_x*x_1 }[/math].

The potential difference between B and C is [math]\displaystyle{ dV = V_C - V_B = -\left(E_x*0 + 0*\left(-y_1 - 0\right) + 0*0\right) = 0 }[/math].

Therefore, the potential difference A and C is [math]\displaystyle{ V_C - V_A = \left(V_C - V_B\right) + \left(V_B - V_A\right) = E_x*x_1 }[/math], which is the same answer that we got when we did not use location B.

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